GLOBAL CURRENCY MOVEMENT AND GENERALIZED RULE INDUCTION

Transcription

1 GLOBAL CURRENCY MOVEMENT AND GENERALIZED RULE INDUCTION A. G. Malliaris Loyola University Chicago, 1 E Pearson, Chicago IL 60611, , tmallia@luc.edu M. E. Malliaris Loyola University Chicago, 1 E Pearson, Chicago IL 60611, , mmallia@luc.edu ABSTRACT This paper uses one form of association analysis, Generalized Rule Induction (GRI), to investigate the relationships among directional movement of eight major currencies priced relative to the dollar. Data over ten years was studied to see if there are movement rules among these currencies that might be stable over time. Eighteen rules were discovered and discussed that appear to have some robustness over both the training and validation sets. For example, when the Australian Dollar and Japanese Yen move in a similar direction, the Euro frequently does the same. These results suggest that co-movement among some specific markets exists over relatively long periods of time and does not exist among others. Keywords: generalized rule induction, foreign exchange, currencies, data mining 1. INTRODUCTION In the search for diversity of assets in which to place funds, an investor prefers to have some investments that do not move together. That is, they do not go up and down simultaneously. However, if it is possible to isolate some assets that do have similar patterns of movement, then this knowledge can be used in two ways: to make money if one has moved and the other has not moved yet, or, to build a more diverse portfolio by not including both in the basket. Similarly, if no pattern of co-movement can be found over time, then the investor is more assured that a pair of specific markets move in a less synchronized fashion. This paper investigates ten years of currency price data movement, using a methodology called Generalized Rule Induction (GRI). GRI was created in 1992 by Smyth and Goodman [13] as an algorithm that could be used for the induction of rules from a set of examples. Rather than using an expert (common, but labor intensive, at that time) to obtain rules, they wanted to create an algorithm that could automatically acquire rules from data, where that data existed. They also wanted their technique to generate rules relating not only two columns of data, but possibly multiple columns. Their focus was on reducing a large dataset to a small set of rules found within that data. In today s internet environment, the acquisition of data is much easier that when this technique was first envisioned. With such ease of data attainment, the number of rule induction methods and problems for their application has increased. These methods continue to be popular for approaching problems in finance, for example, Batyrshin et al [2] [3] use time series databases, Albanis and Batchelor [1] employ rule induction to look at stock selection, Wang et al [15] study

2 forecasting time series, and Tseng [14] discusses co-movement in international stocks and Bossomaier et al [5] detail a simulation of trust in wealth management. This research uses rule induction to look at international currency market data over a ten-year period. In today s world, since it is as easy to trade globally as to trade locally, the growth in the size and complexity of international financial markets has been one of the most striking aspects of the world economy over the last decade. This process of financial globalization and its possible effects has been described by Lane and Milesi-Ferretti [11] [12] and Devereux and Sutherland [9]. Campbell et al.[8] finds that the Australian dollar and the Canadian dollar are positively correlated with local-currency returns on equity markets around the world, including their own domestic markets. At the other extreme, the euro and the Swiss franc are negatively correlated with world stock returns and their own domestic stock returns. The Japanese yen, the British pound, and the US dollar fall in the middle, with the yen and the pound more similar to the Australian and Canadian dollars, and the US dollar more similar to the euro and the Swiss franc. With the increasing globalization, certain currencies have received great significance the past few decades. This global significance translates into a search for the pricing of these currencies. The challenge becomes even greater since currencies are priced one in terms of another. One may view the issue of pricing currencies as a comparison of all economic and financial fundamentals between two nations. Empirical evidence of chaotic dynamics in financial data such as stock market indexes, foreign currencies, macroeconomic time series and several others have been performed by various researchers such as Brock, Scheinkman, Dechert and LeBaron, [7] and Brock and Malliaris[6]. However, there is very little empirical work done to study nonlinear chaotic determinism in currency markets. One way of looking at the interplay of currencies by themselves is to study the direction of movement in each market per day. That is, are there certain major markets that move up or down together sufficiently often for us to form a conclusion about their inter-relationships? This paper uses Generalized Rule Induction to investigate the relationships among directional movement of eight major currencies around the globe. Data over ten years was studied to see if there are movement rules among these currencies that might be stable over time. 2. DATA AND METHODOLOGY We began with daily cash closing prices for the Australian Dollar, British Pound, Brazilian Real, Canadian Dollar, Euro, Japanese Yen, Mexican Peso, and Swiss Franc with respect to the US Dollar. That is, the data reflects the amount of each foreign currency that could be purchased with 1 US dollar that day. Though the foreign exchange market is considered to be a 24-hour market, closing prices can be quoted for individual markets in pairs. Thus, when the market in Tokyo closes, the value of the Yen to the Dollar can be established for that day. All values in this data set are in these units of the foreign currency per US dollar. The data sample covers the time period from January 2000 through July 2009 and was downloaded from Bloomberg. The relative movement in these currencies can be seen in Figure 1. In order to view them all in a similar scale, the Mexican Peso has been multiplied by 10 and the Japanese Yen by 100 for the

3 graph. There are a total of 2,491 observations for prices for each of the eight daily closing prices. These prices were split into two disjoint sets for training and validation. Data from January to June was used as the training set (2215 rows), with the remainder, from July to July , used as the validation set (276 rows). To study the simultaneous market movements, all data was transformed into Up or Down by comparing the value of the currency at time t with its value on the previous day. FIGURE 1. Currency Prices in units of 1 US Dollar, Peso & Yen scaled Association analysis is a popular data mining method that originated with the study of market baskets to see which items people purchased at the same time. It is often used as an exploratory method to help discover interesting relationships in the data that you may wish to analyze further. For an in-depth discussion of association analysis techniques, see Hand et al [10], or Berry and Linoff [4]. The variables in an association analysis model can be specified for use as inputs, outputs, or both. Association analysis then generates a set of rules of the form IF A THEN B where variables specified as inputs may occur after IF, variables specified as outputs may occur after THEN, and variables specified as both may occur in either position. The set of rules that is generated also depend on the user-supplied minimum values of support and confidence. Support refers to the percent of times that some combination of inputs (also called antecedents) occurs in the data set. When the antecedent combination does occur, confidence reflects the percent of time that the output, or consequent, is also true. There are several major association analysis techniques, for example, Apriori, Generalized Rule Induction

4 (GRI), and Carma. These vary in the way they search for interesting rules within a large, and generally sparse, data set. Generalized Rule Induction is a methodology that was introduced in 1992 by Smyth and Goodman [13] when they proposed a new method of automated rule acquisition based on large amounts of data. This methodology introduced a measure of information content called the J- measure. GRI uses the J-measure to quantify the amount of information in a rule, and then generates a set of optimal rules on the training data. The J-measure is defined as J = p(x) [ p(y x) ln (p(y x)/p(y)) + (1 p(y x)) ln (1 p(y x))/(1-p(y)) ] (1) where p(x) is the probability of the observed value of x and p(y) is the probability of y. After running the GRI methodology, a model is generated that consists of a set of rules containing the most information. Along with the statement of each rule, GRI also specifies the specific support and confidence that occurred for each rule. Only rules that meet the userspecified minimum levels for support and confidence are listed. With the easy use of computers today to isolate conditions meeting the antecedents, the level of support (that is, the percent of occurrence within the database) is of less concern that the level of confidence. Thus, if a set of antecedent conditions occurs a very small number of times, but the consequent confidence is high, then this would be a rule worth exploring. 3. PROBLEM SETUP AND RESULTS Association analysis problems most often use data in symbolic format. Thus, all data was converted from numeric values, reflecting their amount relative to 1 US dollar, into non-numeric data indicating only the direction of movement relative to the US dollar on any given day. The Generalized Rule Induction methodology was run using the SPSS product Clementine (now renamed IBM s SPSS Modeler since the purchase of SPSS by IBM). There were two runs of the model for this study with the inputs and outputs divided into groups determined by time zone locations. In the first run of GRI, because of the time difference in markets across the globe, the Australian dollar and Japanese Yen were used as inputs with the Euro, the Swiss Franc and the British Pound as possible outputs. Following this run, the Australian dollar, Japanese Yen, the Euro, the Swiss Franc and the British Pound were used as inputs, with the Mexican Peso, the Brazilian Real and the Canadian dollar as outputs. Only the training set data was used in the generation of the rule sets. Then the validation data was run through the trained rule set to see whether the rules were stable over time. The GRI models generated a total of 151 rules. Here we show a selection of 18 of those rules where the validation set results supported the training set results, sorted by the market of the consequent. Each rule displays the antecedent and consequent followed by the support and confidence in each of the data sets. Rules 1 2, and 3 have, as a consequent, the Brazilian Real. Looking at the first rule, we see that, if both the Australian Dollar and the Euro were down at closing, then, on the same day, the

5 Brazilian Real was down at closing. This simultaneous downward movement in Australia and the Euro occurred in 38.06% of the training set cases. When that simultaneous movement occurred, then 61.09% of the time, the Real was also down in the training set. The support and confidence values are also given for the validation set. In the validation set, we see that the simultaneous down movement for Australia and the Euro occurred in about the same percentage of rows, but the Real s co-movement in the down direction occurred even more often, 73.08% of the time. Rule 2 has the same consequent but an antecedent with movement only from Australia specified. In this rule, we notice that the support is greater than in rule 1, but the confidence is lower. That is, fewer restrictions on the antecedent mean that more rows will display this antecedent, but the percentage of those rows displaying the consequent will be smaller. However, both rules have a high enough confidence to expect that the down movement in the Real is closely tied to that of the Australian Dollar. Rule 3 focuses on conditions necessary for the Real to move up relative to the US dollar. In this case, movements in three markets must happen before a strong consequent appears. Thus, Australia must be up relative to the dollar, the Swiss France must move down, and the Yen must also move down. Rules 4 and 5 have the British Pound as a consequent. These rules indicate that movement in the British Pound, either up or down, is tied to simultaneous co-movements in both Australia and Japan. The confidence levels in both the training and validation sets are in the mid-70s, though the occurrence of the antecedents drops from training to validation. The Canadian dollar is the consequent for rules 6 and 7. It moves down when both Australia and the Euro are down, but up movement depends only on Australia. The percent of time that the consequent was true increased in the validation set for both of these rules. The consequent of rule 6 is true for over 82% of the cases in the validation set. The direction of the Euro is the focus of rules 8 and 9. These show that the movement of the Euro is tied to having movement in both Australia and Japan go in the same direction. That is, all three go up and all three go down together about 80% of the time. These rules did not diminish in confidence from the training to the validation set. Rules 10 through 16 all have the Mexican Peso as a consequent. For the Mexican Peso to move down, either Australia or the Euro must also be down, and some other market must move up, as we see in rules 10 through 13. Up movement in the Peso happens often when either the Euro or Australia are up and the Swiss Franc is down. These rules have much less confidence in the training set than in the validation set, an indication that there may be less robustness in these rules over the time period studied. Finally, rules 17 and 18 indicate that simultaneous movement in Australia and Japan are an indication that the Swiss Franc will go the same direction that day. This relationship holds for both the up and down directions with high confidence. As we see in the table, over 80% of the time, this simultaneous movement occurs.

6 In addition to rule generation using GRI, Clementine allows us to generate a picture of pair-wise relationships among non-numeric data variables using a web graph. This graph, shown in Figure 2, illustrates pair-wise down movements on the left and pair-wise up movements on the right in the training data. Stronger relationships, that is, ones that occur more often, have darker lines connecting them. This graph does not combine up and down movement in the various markets as we can do with GRI, but it does give us some indications of places we might expect rules to be generated. By looking at these graphs, we see that there are more relationships shown for up than for down. Also note that the relationships between the Euro and the Swiss Franc are the strongest in both directions of movement. This figure also enables us to see the power of the GRI methodology over a simple graphic view of a data set. For example, while the links of the Australian dollar and the Japanese Yen to the Swiss Franc are thin on the Up chart here, GRI has shown us that the combination of up values on these two are a powerful indicator of movement in the direction of the Franc as priced in dollars. FIGURE 2. Clementine Web-graph of pair-wise down and up movements

7 TABLE 1. Rules generated by the GRI algorithm RULE TRAINING VALIDATION Antecedent Consequent Sup Conf Sup Conf 1 Australia= Down and Euro= Down Brazil = Down Australia = Down Brazil= Down Australia = Up and Swiss = Down and Japan = Down Brazil = Up Australia = Down and Japan = Down Britian = Down Australia = Up and Japan = Up Britian = Up Australia = Down and Euro = Down Canada = Down Australia = Up Canada = Up Australia = Down and Japan = Down Euro = Down Australia = Up and Japan = Up Euro = Up Australia = Down and Euro = Down and Swiss = Up Mexico = Down Australia = Down and Britian = Down and Japan= Up Mexico = Down Euro = Down and Swiss = Up Mexico = Down Euro = Down and Britian = Down and Swiss= Up Mexico = Down Euro = Up and Britian = Up and Swiss = Down Mexico = Up Euro = Up and Swiss = Down Mexico = Up Australia = Up and Swiss= Down Mexico = Up Australia= Down and Japan = Down Swiss= Down Australia = Up and Japan = Up Swiss = Up CONCLUSIONS For many centuries, national economies have been linked to one another financially, primarily because of trade. The importing nation received goods and paid in some pre-agreed currency. Currency trading predates both bond and stock trading as a financial innovation. However, there is very little doubt that during the past 50 years globalization grew at a remarkable pace and with it, currency trading.

8 Today, the daily volume of currency transactions in currency futures, forwards, swaps and options dominates all other types of trading volumes. This volume is driven by globalization that includes both trade and foreign direct investments, by portfolio diversification and by hedging and speculation, among other factors. Whether currencies move together or independently is a matter of importance for investors wishing to spread the impact of their portfolio decisions. This study used a set of data spanning almost ten years with data reflecting the values of eight major currencies relative to the US dollar. The data was transformed into directional movement and then analyzed to see whether association analysis could uncover rules relating to simultaneous currency movement that remained stable over time. Eighteen rules were discovered and discussed that appear to have some robustness over both the training and validation sets. These results suggest that there is reason to believe that comovement among some specific markets exists over relatively long periods of time and does not exist among others. This information can be used to help diversify a portfolio by holding currencies for which related movements do not occur. 5. REFERENCES [1] Albanis, G. and Batchelor, R. Combining heterogeneous classifiers for stock selection. Intelligent Systems in Accounting, Finance & Management, 2007, 15: doi: /isaf.282 [2] Batyrshin, I., Herrera-Avelar, R., Sheremetov, L, and Panova, A. Association networks in time series data mining, Proceedings of the Fuzzy Information Processing Society, 2005, pgs [3] Batyrshin, I., Herrera-Avelar, R., Sheremetov, L, and Panova, A. Moving Approximation Transform and Local Trend Associations in Time Series Data Bases. Perception-based Data Mining and Decision Making in Economics and Finance, Studies in Computational Intelligence, 2007, Springer, Berlin / Heidelberg. [4] Berry, M. and Linoff, G. Data Mining Techniques, Second Edition, Wiley Publishing Inc., [5] Bossomaier, T., Standish, R., Harre, M. Simulation of trust in client-wealth management adviser relationships, International Journal of Simulation and Process Modelling, 2010, 6(1), [6] Brock, W., and Malliaris, A.. Differential equations, stability and chaos in dynamic economics. North Holland, Amsterdam, The Netherlands, [7] Brock, W., Scheinkman, J., Dechert, W. and LeBaron, B., A test for independence based on the correlation dimension, Econometric Reviews, 1996, Taylor and Francis Journals, 15(3),

9 [8] Campbell, J.Y., Medeiros, K.S., and Viceira, L.M., Global Currency Hedging, NBER Working Paper, No. W13088, [9] Devereux, M.B., and Sutherland, A., Financial Globalization and Monetary Policy, IMF Working Paper, [10] Hand, D., Mannila, H., and Smyth, P. Principles of Data Mining, The MIT Press, [11] Lane, P., and Milesi-Ferretti, G.M., The External Wealth of Nations: Measures of Foreign Assets and Liabilities for Industrial and Developing Countries, Journal of International Economics, 2001, 55, [12] Lane, P., and Milesi-Ferretti, G.M., The External Wealth of Nations Mark II, IMF Working Paper, No 06-69, [13] Smyth P, Goodman RM. An information theoretic approach to rule induction from databases. IEEE Transactions on Knowledge and Data Engineering. 1992;4(4): doi: / [14] Tseng, C. Data driven modeling of co-movement among international stock market, Journal of Modelling in Management, 2007, 2(3), [15] Wang, X., Smith-Miles, K., Hyndman, R., Rule induction for forecasting method selection: Meta-learning the characteristics of univariate time series, Neurocomputing, June 2009, 72(10-12),